For the pork, preheat the oven to 325 degrees . Pat the pork dry with paper towels, then season generously with kosher salt and pepper. In a large Dutch oven, heat the oil, two turns of the pan, over medium-high. Add the pork and cook until browned on all sides, about 12 minutes. Transfer the pork to a plate.

Reduce the heat to medium. Add the next 6 ingredients to the pot and season. Cook, stirring often, until the vegetables soften, about 10 minutes. Add the beer and stir, scraping up browned bits. Let simmer for 1 minute. Add the chicken stock and the pork with any juices. Increase the heat to medium-high and bring to a gentle boil.

Cover the pot and transfer to the oven. Cook, turning the pork halfway through, until the meat is tender and falls apart when pulled with a fork, about 2 hours and 15 minutes. Remove twine, if using. Transfer the pork to a platter and let cool slightly. Set a sieve over a large bowl and strain broth. Using 2 forks, shred the pork and transfer to a large bowl.

While the pork is cooking, make the sauce: In a small saucepan, bring the chiles and 2 cups water to a boil. Reduce the heat to medium and simmer until tender, about 10 minutes. Transfer the mixture to a blender and mix until smooth.

Meanwhile, in a medium saucepan, bring the ketchup and the next 6 ingredients to a simmer over medium heat. Add the chile puree and season with pepper. Simmer, stirring occasionally, until flavors meld, about 15 minutes. Stir in the lime juice. Add the BBQ sauce to the pork and enough reserved cooking broth (about 2 cups) to keep moist.